I believe it is a fixed phrase (‘injury problems’ does not work as a substitute) and can admit that there is no account of a singular form (‘injury woe’). So, it makes the said phrase an idiom.

I would like to clarify its meaning and usage. It seems that the idiom is limited to sports and is found mainly in news. Upon checking the web for contexts, it might be used to describe a series of injuries afflicting a player or several injuries in a row for different players.

A First Division football team manager, for example, could feasibly say something like Despite our injury woes last season, we still managed to avoid relegation to the Second Division. Where "injury woes" means the trials and tribulations we suffered because of injuries (to our players). It's a rather stylised turn of phrase - but not really an "idiom", any more than our injury problems, our injury setbacks,...
– FumbleFingersOct 10 '18 at 15:16

Injury woes could indeed be coined injury problems or injury setbacks as stated by Fumble Fingers in the comment in reply to OP's question.

The phrase is mainly used in sport as mentioned by OP, particularly in football. It refers to the difficulty teams have in creating optimized player squads because of the number of unavailable players recovering from their injuries.

is confident injuries to Mousa Dembele and Erik Lamela are minor amid fears of a crisis in his midfield.

Central midfielders, Eric Dier and Dembele, have been vacationing during Spurs’ pre-season tour in America due to their World Cup exploits, while Harry Winks and Josh Onomah stayed at home to nurse injuries.

Victor Wanyama, Moussa Sissoko and 18-year-old Tashan Oakley-Boothe have all sustained injuries during Spurs’ preparations ahead of the new season, which could potentially leave Pochettino without seven central midfielders when they face Newcastle in their opening game of the 2018-19 Premier League season on 11 August.

“We’ve already got Bongani Zungu and Themba Zwane out‚ we’ve got Keagan Dolly out‚ and then you bring in a Morena who I thought really was perfect for this game. And then he pulls a hamstring‚” Baxter said after training Steyn City School in Fourways, Johannesburg.

“And then you’ve also got Vincent Pule‚ who’s got a bad one. The scans showed that he’s got damage on is cruciate ligament. So it isn’t just a kick‚ as people might have first thought.

“Those were two withdrawals that caused a little bit of concern.

“But the lads we’ve brought in – Mkhize will arrive later‚ and Lorch is here – will add cover.

“The other one not here is Dino Ndlovu. His flight was cancelled in Hong Kong. He very professionally sent me a video of the board where it was actually saying in Chinese‚ and then it flicked over‚ that he was delayed.

“So that’s not ideal either – he’s only going to have two days’ preparation.